# Electrical Forces Discrepancy

#### PlasMav

Hello,
First of all I want to say that I am not an EE, I am an ME but I am working for an EE laboratory on campus and am trying to get a force from and electrical circuit on a project here.

I am using the following equations:

F = 1/2*L'*I^2 (where L' = H/m)

and

F = u_0*I_1*I_2*d/(2*pi*r) (where d = length of conductors and r = their separation distance)

I am trying to get the forces acting upon two parallel conductors with opposite current direction. With my hand calculations and simulation in LTspice I am getting a number greater by a factor of 5 with the second equation than with the first.
I have reason to believe the first equation is not valid for what I want but I want to get a second or third opinion.

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#### berkeman

Mentor
F = 1/2*L'*I^2 (where L' = H/m)

and

F = u_0*I_1*I_2*d/(2*pi*r) (where d = length of conductors and r = their separation distance)
Can you post links to where you got those two equations?

For one thing, the units of the first equation look wrong to me. The RHS has units of energy, not force...

#### PlasMav

Can you post links to where you got those two equations?

For one thing, the units of the first equation look wrong to me. The RHS has units of energy, not force...

View attachment 249159

The equation I am using comes from that equation. The difference is L and L'

The L in the energy equation is inductance (Henry) and the L' in the force equation I am using is inductance/distance (Henry/meter).

So since Work or Energy = Force * Distance, Energy/Distance = Force.

Does this make more sense?

The first equation comes from: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/wirfor.html

#### Dr.D

I think that the problem is a misinterpretation of L'. If we denote the separation distance as s (not the d used by the OP), then L'=dL/ds. This is the result obtained if you apply the principle of Virtual Work to the situation. An ME should especially appreciate this approach (yes, I'm an ME also).

#### JBA

Gold Member
For an alternative to compare with your results use the force calculator at the below website:

PS I am also an ME and get irritated when formulas are given with symbols that are not clearly defined because the author, believes they are obvious to the user.

Edit

#### Dr.D

PS I am also an ME and get irritated when formulas are given with symbols that are not clearly defined because the author, believes they are obvious to the user.
I think the symbols are defined, provided you subscribe to the common notation that a prime (') represents a derivative. Of course, dL/ds would be more obvious, but it takes more time to write and most of us don't want to invest that time. It all depends on where the individual is coming from.

#### JBA

Gold Member
I think the symbols are defined, provided you subscribe to the common notation that a prime (') represents a derivative. Of course, dL/ds would be more obvious, but it takes more time to write and most of us don't want to invest that time. It all depends on where the individual is coming from.
I understand, and need to clarify my comment because it was not related to any of the information or formulas posted in this thread.
My comment was the result of my frustration during a search of the web to find information on this subject. There were a number of referenced sites that discussed all of the elements of the issue and included a number of formulas and derivations comprised totally of symbols with no definitions their units. I understand the brevity in our forums' posts, but I expect the symbols in equations in published papers and references to be clearly defined.

#### PlasMav

My apologies. I believe L' was not meant to be a derivative, just another variable separate from inductance. View it as a 'distributed inductance' along the conductor. For any length of the conductor, there is a certain inductance per meter. This is what L' is. The separation distance was not involved in this equation, only the other equation.

This notation was not my own doing, it was given to me as such.

#### tech99

Gold Member
F = u_0*I_1*I_2*d/(2*pi*r) (where d = length of conductors and r = their separation distance)
I think this one is approximately correct but I struggle with the symbols.
Field of an infinitely long wire = (mu I) /(2 pi D), where for mu I mean the permeability of free space.
Force on a wire in a given field = I L B
Force = I L mu I / 2 pi D = mu I^2 L / 2 pi D (approx)

Note: it looks as if you found formulas using L to mean length and also inductance.

Last edited:

#### PlasMav

I think this one is approximately correct but I struggle with the symbols.
Field of an infinitely long wire = (mu I) /(2 pi D), where for mu I mean the permeability of free space.
Force on a wire in a given field = I L B
Force = I L mu I / 2 pi D = mu I^2 L / 2 pi D (approx)

Note: it looks as if you found formulas using L to mean length and also inductance.
That is from here

Each formula is from a different place. The only length I have is 'd' in the second formula. L' is inductance per length.

#### PlasMav

So which force is most valid for lateral force between conductors?

In my opinion the first equation it is still possible[except the factor 1/2!].

Instead of ΔL I put "lng" in order to avoid confusion. Then the force formula it is:

F = μo*I1*I2*lng/(2*pi*r) or:

F = μo*I1/(2*pi*r)*I2*lng but:

B(r)=μo*I1/2πr then:

F =B(r)*I2*lng

B(x)=μo*I1/2πx magnetic flux density at distance x from the first conductor center.

Φ(r)=ʃB(x)*lng*dx|x=ro to r] where ro it is conductor radius and r it is distance center-line to

center line.

L(r)=Φ(r)/I1

dL(r)/dr=B(r)*lng/I1 then:

B(r)=dL(r)/dr/lng*I1
and substituting in above formula we get:

F=I2*dL(r)/dr*I1

If L(r)=μo/2π*lng*LN(r/ro) where ro it is the bare conductor radius or:

L(r)=μo/2π*lng*[LN(r)-LN(ro)] then

dL(r)/dr=μo.lng/2πr

F=μo.lng/2πr*I1*I2 the same formula as above.

"Electrical Forces Discrepancy"

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